A137967 - OEIS

%I #10 Mar 03 2018 13:53:42

%S 1,1,2,13,66,406,2602,17271,118444,829514,5914980,42791085,313277294,

%T 2316793170,17281455882,129867946828,982293317064,7472406051744,

%U 57132051350160,438797394096378,3383870656327576,26191385476141936

%N G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^6)^2.

%H Vaclav Kotesovec, <a href="/A137967/b137967.txt">Table of n, a(n) for n = 0..400</a>

%F G.f.: A(x) = 1 + x*B(x)^2 where B(x) is the g.f. of A137968.

%F a(n) = Sum_{k=0..n-1} C(2*(n-k),k)/(n-k) * C(6*k,n-k-1) for n>0 with a(0)=1. - _Paul D. Hanna_, Jun 16 2009

%F a(n) ~ sqrt(2*s*(1-s)*(6-7*s) / ((132*s - 120)*Pi)) / (n^(3/2) * r^n), where r = 0.1201742080825038015263858974579392344239858277873... and s = 1.297009871974239150024579315539982910111693413337... are real roots of the system of equations s = 1 + r*(1 + r*s^6)^2, 12 * r^2 * s^5 * (1 + r*s^6) = 1. - _Vaclav Kotesovec_, Nov 22 2017

%o (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^6)^2);polcoeff(A,n)}

%o (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(2*(n-k),k)/(n-k)*binomial(6*k,n-k-1))) \\ _Paul D. Hanna_, Jun 16 2009

%Y Cf. A137968, A137966; A137952, A137955, A137960.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 26 2008